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Traditional performance metrics, usually based on linear factor models, do not accurately measure the efficiency of hedge funds. Prof. Paul Karehnke and his co-author propose a more flexible model that does so by considering skewness to better guide portfolio choices – in particular to inform diversification strategies.

The beauty of linear factor models is that they neatly break down relations into explainable (and less explainable) parts, creating patterns that are easy to catch at a glance. It's one of the reasons performance evaluation of investment funds is commonly based on such models, which help weigh risks against expected returns of investment portfolios – usually viewed relatively to a benchmark. The general assumption guiding fund managers that use these mean-variance models is that investors are risk-averse. Except that these linear models, such as the capital asset pricing model (CAPM), the Fama-French three-factor model or the Fama-French-Carhart four-factor model, have a major shortcoming: “they fail to capture nonlinearities in returns that arise from trading strategies involving dynamic trading, derivative usage and leverage. It's particularly true of hedge funds, which usually have freer rein than mutual funds to invest aggressively and offer option-like payoffs,” explains professor Paul Karehnke. This means distribution patterns for alpha (an investment strategy's ability to beat the market, or its "edge") do not take on the familiar bell-curve shape and this asymmetry, or skewness, can blindside investors. “Even approaches that attempt to capture nonlinearities by including factors with nonlinear returns, such as option payoffs, next to the standard factors are insufficient,” he adds. Indeed, option-like payoffs can still generate a positive alpha at the expense of negative skewness (when distribution is impacted by negative outliers).

Improving spanning tests of portfolios

With Tilburg University’s Frans de Roon, Paul Karehnke proposes to remedy the shortcomings of linear models by accounting for the skewness preference of investors in performance evaluation. Expanding on Huberman and Kandel's 1987 work, they developed a spanning test for assets with option-like payoffs to assess whether they would improve an investment opportunity set. Such general frameworks are a modern twist on the question of whether to put all your eggs in the same basket (or perhaps more accurately whether to add different kinds of eggs to your basket): Will including additional assets lower the total risk of investing without sacrificing returns? “Our framework is particularly relevant to examine hedge fund returns compared with mutual fund returns,” he claims.

Considering an investment opportunity set of benchmark assets versus a larger set of benchmark plus additional assets, they proposed two tests for whether skewness investors (which we will use as shorthand for “risk-averse mean-variance-skewness investors”) benefit from additional assets. First, a spanning test considers the hypothesis that no investor benefits from the additional assets versus the alternative in which at least one investor benefits – they refer to “one investor” as a particular combination of preferences over mean, variance and skewness. Second, an overlap test considers the hypothesis that at least one investor does not benefit versus the alternative, that all investors do. This overlap concept generalizes the mean-variance intersection test, testing for returns for a group of investors.

Refining strategies for “skewness investors”

They applied their two tests using data from a sample of more than 4,700 live and dead hedge and mutual funds from Morningstar, a global investment research firm, covering the period 1994-2014. Our analysis took the perspective of investors who could invest in bonds, stocks, and a risk-free asset (the latter for the sake of clarity of discussion, since risk-free assets are not exactly realistic in modern financial markets). The proxy for stocks and bonds were the S&P 500 Total Return Index from Morningstar and the 10-year US Treasury Bond Index from the Center for Research in Security Prices. “Considering investors who currently invest in stocks and bonds, we obtained different results with our method compared with standard spanning tests: among the funds that benefit all mean-variance investors, 73% still do so for classic linear models, but only 15% improve the investment opportunity set of all skewness investors,” states Paul Karehnke. “Thus, the majority of the hedge funds that are attractive from a mean-variance point of view have a trade-off with negative coskewness with stocks and bonds that do not make them attractive for groups of mean-variance-skewness investors. More generally, all skewness investors improve their investment opportunity set with around 11% of the hedge funds, but with less than 4% of the mutual funds.”
To fine-tune their results, they took a closer look at hedge fund characteristics and strategies. “Our tests suggest there is strong evidence against spanning with short sales, and slightly weaker evidence without short sales. So only portfolios of global derivatives (systematic futures strategies being the most beneficial) seem to benefit all skewness investors as they have desirable positive coskewness, as opposed to event and relative value funds that mostly follow arbitrage strategies and have more negative coskewness,” he adds.
They also find that hedge funds are better able to actively vary their exposure to market and macroeconomic risk in an advantageous way than mutual funds, very few of which have macro timing abilities. “As a result, hedge funds generate both positive alpha and desirable coskewness and are more likely to be attractive to all skewness investors.”

“Our results show that taking into account skewness in performance evaluation has significant economic value for investors,” he concludes. This method is applicable to other asset classes, too. “In particular, the returns of currency-trading strategies and emerging markets have also exhibited skewness, making them potentially good application fields for our model.”

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